Usually no, with four exceptions: COBRA, coverage while on unemployment, long-term care insurance, and Medicare premiums after 65.
Ordinary health insurance premiums are not qualified expenses. Four exceptions are:
Medicare supplement (Medigap) policies do not qualify, which is a common and expensive mix-up.
The Medicare exception is a genuinely large benefit. Part B alone runs well over $2,000 a year per person, and paying it from an HSA converts that into a tax-free expense for the rest of your life.
Note the asymmetry after 65: you can pay Medicare premiums from an HSA, but you can no longer contribute once you enroll in Medicare.